It is clear that strong-efficiency implies efficiency. This notion sets the bar too high on SCCs, since there are no rules satisfying it on unrestricted domains. This problem arises because Condorcet cycles can arise in those domains. But even efficiency is a very demanding condition:The definitions of the voting rules as well as the proof of Proposition REF  can be found in the Appendix. The definitions may also be found in [1]} and [2]}.
